delid wrote:

Bunuel wrote:

vitorpteixeira wrote:

If |x–1|<4, so:

(A) x > 3

(B) x ≤ 4

(C) -4< x < 4

(D) x > -4

(E) none of the above

\(|x–1|<4\);

\(-4<x-1<4\);

\(-3<x<5\).

Any x from that range will be more than -4.

Answer: D.

P.S. What is the source of this question?

Hi

Bunuel, thanks for the explanation. I tend to approach this questions in the wrong way, trying to find an answer choice that matches with the range found through the inequalities. For instance, in this case I was looking for -3<x<5 (or something more restrictive, such as -2 < x < -5), which led me to pick answer E.

Given your explanation, I understand that I should look for that condition that holds for every x in the range...but I still have a huge doubt.

You say it is (D) x > -4 because any X from that range will be more than -4. Following your logic, wouldn't also B) x ≤ 4 be true? Our range is -3<x<5, which means that every x will be less or equal to 4.

Is there something I am not getting right?

Your help is super appreciated

Although this is a relatively simple question on absolute values (modulus), the way in which the options have been framed can put you off.

The reason for this is that, you go in to the solution, expecting to see a range that you think you’ll get when you solve the given inequality. But, in reality, none of the options represent the exact range that you will get when you solve the inequality.

Therefore, it becomes important to understand that , here, we are trying to find out a range of values, which in turn fully covers the range which satisfies the given inequality.

If |x-a| < y, the range of x that satisfies the inequality will be a-y < x < a+y. When we apply this to the given inequality, |x-1| < 4, we can say that -3<x<5. Let us represent this on a number line, as shown below:

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21st June 2019 - Reply 3 - 1.JPG [ 14.93 KiB | Viewed 989 times ]
Let us evaluate the options now.

Option 1 says x>3. This means that x can be any value from 3 to infinity. Clearly, only a portion of this range satisfies our inequality i.e. 3<x<5.

So, quite obviously, you cannot say that all values which satisfy |x-1|<4 fall in the range of x>3.

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21st June 2019 - Reply 3 - 2.JPG [ 16.29 KiB | Viewed 990 times ]
A similar thing works against option B.

With option C, although some numbers are within the range, some numbers aren’t. Like for example, x = 4.5 satisfies the given inequality but is not within -4<x<4.

When it comes to option D, we can say that all the values that satisfy our inequality are greater than -4. This means, when I pick any value from the range satisfying the inequality, it will always qualify as true when compared with option D.

The same cannot be said about the other options, as we have demonstrated.

So, the correct answer is D.

The confusion that this question has created could have been nullified if the question was framed as a ‘Must be’ question. For example “ If |x-1|<4, which of the following must be true for the values of x that satisfy the inequality?” . Essentially, this is what the question is testing you on.

Hope this helps!

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